let A, B be non empty transitive with_units AltCatStr ; :: thesis: for F being reflexive feasible FunctorStr over A,B st F is bijective & F is comp-preserving & F is Covariant & F is coreflexive holds
F " is comp-preserving
let F be reflexive feasible FunctorStr over A,B; :: thesis: ( F is bijective & F is comp-preserving & F is Covariant & F is coreflexive implies F " is comp-preserving )
assume A1: ( F is bijective & F is comp-preserving & F is Covariant & F is coreflexive ) ; :: thesis: F " is comp-preserving
set G = F " ;
A2: F " is Covariant by A1, Th38;
reconsider H = F " as reflexive feasible FunctorStr over B,A by A1, Th35, Th36;
A3: the ObjectMap of (F ") = the ObjectMap of F " by A1, Def38;
consider ff being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F such that
A4: ff = the MorphMap of F and
A5: the MorphMap of (F ") = (ff "") * ( the ObjectMap of F ") by A1, Def38;
A6: F is injective by A1;
then F is one-to-one ;
then A7: the ObjectMap of F is one-to-one ;
F is faithful by A6;
then A8: the MorphMap of F is "1-1" ;
F is surjective by A1;
then F is full ;
then A9: ex f being ManySortedFunction of the Arrows of A, the Arrows of B * the ObjectMap of F st
( f = the MorphMap of F & f is "onto" ) ;
let o1, o2, o3 be Object of B; :: according to FUNCTOR0:def 21 :: thesis: ( <^o1,o2^> <> {} & <^o2,o3^> <> {} implies for f being Morphism of o1,o2
for g being Morphism of o2,o3 ex f9 being Morphism of ((F ") . o1),((F ") . o2) ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 ) )
assume A10: <^o1,o2^> <> {} ; :: thesis: ( not <^o2,o3^> <> {} or for f being Morphism of o1,o2
for g being Morphism of o2,o3 ex f9 being Morphism of ((F ") . o1),((F ") . o2) ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 ) )
then A11: <^(H . o1),(H . o2)^> <> {} by A2, Def18;
assume A12: <^o2,o3^> <> {} ; :: thesis: for f being Morphism of o1,o2
for g being Morphism of o2,o3 ex f9 being Morphism of ((F ") . o1),((F ") . o2) ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
then A13: <^(H . o2),(H . o3)^> <> {} by A2, Def18;
A14: <^o1,o3^> <> {} by A10, A12, ALTCAT_1:def 2;
then A15: <^(H . o1),(H . o3)^> <> {} by A2, Def18;
then A16: <^(F . ((F ") . o1)),(F . ((F ") . o3))^> <> {} by A1, Def18;
let f be Morphism of o1,o2; :: thesis: for g being Morphism of o2,o3 ex f9 being Morphism of ((F ") . o1),((F ") . o2) ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
let g be Morphism of o2,o3; :: thesis: ex f9 being Morphism of ((F ") . o1),((F ") . o2) ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
reconsider K = F " as Covariant FunctorStr over B,A by A1, Th38;
K . f = (Morph-Map (K,o1,o2)) . f by A10, A11, Def15;
then reconsider f9 = (Morph-Map (K,o1,o2)) . f as Morphism of ((F ") . o1),((F ") . o2) ;
K . g = (Morph-Map (K,o2,o3)) . g by A12, A13, Def15;
then reconsider g9 = (Morph-Map (K,o2,o3)) . g as Morphism of ((F ") . o2),((F ") . o3) ;
take f9 ; :: thesis: ex g9 being Morphism of ((F ") . o2),((F ") . o3) st
( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
take g9 ; :: thesis: ( f9 = (Morph-Map ((F "),o1,o2)) . f & g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
thus f9 = (Morph-Map ((F "),o1,o2)) . f ; :: thesis: ( g9 = (Morph-Map ((F "),o2,o3)) . g & (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 )
thus g9 = (Morph-Map ((F "),o2,o3)) . g ; :: thesis: (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9
consider f99 being Morphism of (F . ((F ") . o1)),(F . ((F ") . o2)), g99 being Morphism of (F . ((F ") . o2)),(F . ((F ") . o3)) such that
A17: f99 = (Morph-Map (F,((F ") . o1),((F ") . o2))) . f9 and
A18: g99 = (Morph-Map (F,((F ") . o2),((F ") . o3))) . g9 and
A19: (Morph-Map (F,((F ") . o1),((F ") . o3))) . (g9 * f9) = g99 * f99 by A1, A11, A13;
A20: g = g99 by A1, A12, A18, Th40;
A21: f = f99 by A1, A10, A17, Th40;
A22: [((F ") . o1),((F ") . o3)] in [: the carrier of A, the carrier of A:] by ZFMISC_1:87;
A23: [o1,o3] in [: the carrier of B, the carrier of B:] by ZFMISC_1:87;
then A24: [o1,o3] in dom the ObjectMap of (F ") by FUNCT_2:def 1;
dom the MorphMap of F = [: the carrier of A, the carrier of A:] by PARTFUN1:def 2;
then [((F ") . o1),((F ") . o3)] in dom the MorphMap of F by ZFMISC_1:87;
then A25: Morph-Map (F,((F ") . o1),((F ") . o3)) is one-to-one by A8;
[((F ") . o1),((F ") . o3)] in dom the ObjectMap of F by A22, FUNCT_2:def 1;
then A26: ( the Arrows of B * the ObjectMap of F) . [((F ") . o1),((F ") . o3)] = the Arrows of B . ( the ObjectMap of F . (((F ") . o1),((F ") . o3))) by FUNCT_1:13
.= the Arrows of B . ((F . ((F ") . o1)),(F . ((F ") . o3))) by A1, Th22
.= <^(F . ((F ") . o1)),(F . ((F ") . o3))^> by ALTCAT_1:def 1 ;
Morph-Map (F,((F ") . o1),((F ") . o3)) is Function of ( the Arrows of A . [((F ") . o1),((F ") . o3)]),(( the Arrows of B * the ObjectMap of F) . [((F ") . o1),((F ") . o3)]) by A4, A22, PBOOLE:def 15;
then A27: dom (Morph-Map (F,((F ") . o1),((F ") . o3))) = the Arrows of A . (((F ") . o1),((F ") . o3)) by A16, A26, FUNCT_2:def 1
.= <^((F ") . o1),((F ") . o3)^> by ALTCAT_1:def 1 ;
A28: ( the Arrows of A * the ObjectMap of (F ")) . [o1,o3] = the Arrows of A . ( the ObjectMap of H . (o1,o3)) by A24, FUNCT_1:13
.= the Arrows of A . (((F ") . o1),((F ") . o3)) by A2, Th22
.= <^((F ") . o1),((F ") . o3)^> by ALTCAT_1:def 1 ;
the MorphMap of (F ") is ManySortedFunction of the Arrows of B, the Arrows of A * the ObjectMap of (F ") by Def4;
then Morph-Map ((F "),o1,o3) is Function of ( the Arrows of B . [o1,o3]),(( the Arrows of A * the ObjectMap of (F ")) . [o1,o3]) by A23, PBOOLE:def 15;
then A29: dom (Morph-Map ((F "),o1,o3)) = the Arrows of B . (o1,o3) by A15, A28, FUNCT_2:def 1
.= <^o1,o3^> by ALTCAT_1:def 1 ;
A30: Morph-Map ((F "),o1,o3) = (ff "") . ( the ObjectMap of (F ") . (o1,o3)) by A3, A5, A24, FUNCT_1:13
.= (ff "") . [(H . o1),(H . o3)] by A2, Th22
.= (Morph-Map (F,((F ") . o1),((F ") . o3))) " by A4, A8, A9, A22, MSUALG_3:def 4 ;
A31: the ObjectMap of (F * H) = the ObjectMap of F * the ObjectMap of H by Def36
.= the ObjectMap of F * ( the ObjectMap of F ") by A1, Def38
.= id (rng the ObjectMap of F) by A7, FUNCT_1:39
.= id (dom the ObjectMap of (F ")) by A3, A7, FUNCT_1:33
.= id [: the carrier of B, the carrier of B:] by FUNCT_2:def 1 ;
[o1,o1] in [: the carrier of B, the carrier of B:] by ZFMISC_1:87;
then A32: the ObjectMap of (F * H) . (o1,o1) = [o1,o1] by A31, FUNCT_1:18;
A33: F . ((F ") . o1) = (F * H) . o1 by Th33
.= o1 by A32 ;
[o2,o2] in [: the carrier of B, the carrier of B:] by ZFMISC_1:87;
then A34: the ObjectMap of (F * H) . (o2,o2) = [o2,o2] by A31, FUNCT_1:18;
A35: F . ((F ") . o2) = (F * H) . o2 by Th33
.= o2 by A34 ;
[o3,o3] in [: the carrier of B, the carrier of B:] by ZFMISC_1:87;
then A36: the ObjectMap of (F * H) . (o3,o3) = [o3,o3] by A31, FUNCT_1:18;
A37: F . ((F ") . o3) = (F * H) . o3 by Th33
.= o3 by A36 ;
(Morph-Map ((F "),o1,o3)) . (g * f) in rng (Morph-Map ((F "),o1,o3)) by A14, A29, FUNCT_1:def 3;
then A38: (Morph-Map ((F "),o1,o3)) . (g * f) in dom (Morph-Map (F,((F ") . o1),((F ") . o3))) by A25, A30, FUNCT_1:33;
(Morph-Map (F,((F ") . o1),((F ") . o3))) . ((Morph-Map ((F "),o1,o3)) . (g * f)) = (Morph-Map (F,((F ") . o1),((F ") . o3))) . (g9 * f9) by A1, A14, A19, A20, A21, A33, A35, A37, Th40;
hence (Morph-Map ((F "),o1,o3)) . (g * f) = g9 * f9 by A25, A27, A38; :: thesis: verum